Floor plan - Your opinions, ideas, and suggestions

Precisely because of the water. Otherwise, in a cold winter the pipes can burst.

Ok that makes the whole thing a bit more difficult again. Hmm, I will discuss the [Hauswirtschaftsraum] problem with the architect. Maybe he also has an idea of what could be sorted out there well and relatively easily.

Honestly, me

It's not possible anymore. On the one hand because this will be a prefabricated garage and on the other hand financially no longer feasible.

Honestly, it always scares me when a house construction is financed on such a tight budget. All it takes is a little something to go wrong (and I don't know any construction where everything goes smoothly) and it becomes tight.
I wouldn't have that much courage...

I don't know how you come to the conclusion that it is underfinanced. There are different problems:

1) If the garage gets bigger, it won't be so easy with the boundary construction anymore (9m length is the maximum if I'm not mistaken)
2) Since these are prefabricated garages, an extension would then be a special size. I think everyone knows that custom-made items often end up in the five-figure range. And that's not worth it to me for one or two meters more garage

The idea was justified, but I think "just" because a utility room is very small, immediately building a custom-made garage is disproportionate. At least for me ;-)

Without the door to the garage from the utility room, it will be sufficient for the washing machine, vacuum cleaner, etc.

I would reserve the space by the stairs for the urgently needed wardrobe. Think of shoes, handbags, hats, scarves, etc.

either you place the garage on the boundary (boundary construction) or 3 meters away from the neighbor. You will not be allowed 2 meters.

That is not correct!

Buildings with an average wall height of up to 3 m above the ground surface at the boundary, which are used as a garage, greenhouse, or for storage purposes, are permitted without their own clearance areas as well as within the clearance areas of a building
- without openings in the walls facing the neighbor's boundary,

- including subordinate facilities installed on them for solar energy generation and antenna systems each up to 1.5 m in height,

- even if they are not attached to the property boundary or to a building,

- even if the building has access to another building.
Source: LBV NRW, §6, para. 11

Without clearance or with a clearance reduced to 1 m from the boundary, the following are permitted
1. Garages and buildings without habitable rooms and fireplaces with a height of up to 3 m and... Source: LBV Nds 2012 §5, para. 8

Certainly other states have similar passages in their regulations.

The idea was justified, but I think "just" because a utility room is very small, building a custom garage directly is disproportionate.

Of course it is disproportionate, which is why you are also advised to paint over the door.
I wonder anyway how you want to get the sack and the box past the car parked in the garage? There is only one alternative: park outside and use a door. Which one you ultimately use is up to you. But basically your intermediate door will be gone or the procedure will simply be annoying.